Understanding the Single Sample T Test
A single sample t test checks whether one sample mean differs from a fixed benchmark. It is useful when the population standard deviation is unknown. The test estimates uncertainty with the sample standard deviation. It then compares the observed difference with the standard error.
When This Test Helps
Use this method when you have one numerical sample. The observations should be independent. The data should be roughly normal, especially for small samples. Larger samples are more forgiving because the sampling distribution becomes steadier. Common uses include quality checks, class score comparisons, lab measurements, and survey averages.
What the Result Means
The t statistic shows how many standard errors separate the sample mean from the hypothesized mean. A large absolute t value gives stronger evidence against the null hypothesis. The p value measures how unusual the result is when the null hypothesis is true. If the p value is less than alpha, the result is usually called statistically significant.
Confidence and Effect Size
The confidence interval gives a practical range for the true mean. If a two sided interval does not include the hypothesized mean, it matches a two sided significant result at the related alpha level. Cohen's d reports the difference in sample standard deviation units. Hedges' g adds a small sample correction, which is helpful when n is limited.
Good Practice
Always review the data before trusting the test. Look for entry errors, extreme outliers, and strong skew. Report the sample size, mean, standard deviation, t value, degrees of freedom, p value, confidence interval, and effect size. Statistical significance alone is not enough. A small effect may be unimportant in practice. A nonsignificant result may still matter when the sample is too small. Use the decision together with subject knowledge and clear assumptions.
Reading Assumptions
The calculator does not replace judgment. It supports clean reporting and quick checking. For paired observations, use a paired t test instead. For two unrelated groups, use an independent samples test. For proportions, use a proportion test. If the data are ordinal, highly skewed, or filled with outliers, consider a nonparametric method and explain the choice. This keeps the conclusion honest and easier for readers to audit later.